Question

We observe the following US Treasury Bonds with semi-annual coupon payments, each with a face value

of $10000.

A bond maturing in 6 months, with a 3% coupon rate, is trading for $10099.25

A bond maturing in 12 months, with a 5% coupon rate, currently costs $ 10293.75

A bond maturing in 18 months, with a 5% coupon rate, costs $10231.25

A bond maturing in 24 months, with a 4% coupon rate, costs $10071.00

(a) Use these bonds to and the 6, 12, 18, and 24 month discount factors.

(b) What are the spot rates associated with the discount factors from (a)?

(c) You are considering a potential project which costs $900. It makes semi-annual payments for

two years (4 payments). The first one is $230. After that, the cash flow grows at an annualized

rate of 5%. What is the NPV of this project?

Answer 1

Greetings,

A and b) 6 months

PV = 10099.25

FV = 10000+150= 10150

Using simple calculator, discounting factor = 10099.25/10 150= 0.995 So 6m spot rate = (1/0.995)-1*100 = 0.5025%. This is for 6m, so annualised rate is 0.5025*2=1.005%

For 12 months,

Semi annual coupon = 250

PV of first coupon = 250*0.995 = 248.75. Price = 10293.75.

So price net of first coupon = 10293.75 - 248.75 = 10045. This step is known as forward substitution or bootsrapping. Price includes 6m coupon which needs to be pulled at 6m spot rate. Rest whole is attributable to 12m coupon plus FV which will give us 12m rate and discounting factor.

12 month discounting factor = 10045/10250* = 0.98

12 months spot rate = (1/0.98) -1 *100 = 2.041%

* 10000 is face value and 250 is second coupon receivable on 12 months end

For 18 months ,

Semi annual coupon = 250

PV of first coupon = same as above = 248.75

PV of second coupon = 250*0.98=245

Remaining price = 10231.25 - 248.75 - 245 = 9737.50

Amount Receivable on 18 months = 10250

Discounting Factor = 9737.50/10250 = 0.95

18 months rate = ((1/0.95)-1)*100 = 5.263%. This is for 18 months, but rates are always denoted in p.a terms. So p.a. rate = 5.263*12/18 = 3.51%

Note - For simplicity , 6m and 18 m rates have been calculated on bond yield basis ie power form not used, only simple interest calculation is done.

For 24 months,

Semi annual coupon = 200

PV of first coupon = 200*0.995 = 199

PV of second coupon = 200*0.98 = 196

PV of third coupon = 200*0.95 = 190

Remaining Price = 10071 - 199-196-190 =9486

Amount Receivable on maturity = 10200

Discounting Factor = 9486/10200=0.93

24 months rate = ((1/0.93)-1)*100 = 7.5269%. This is 24 months rae , so 12 months rate = 7.5269/2 = 3.76345%

C) We consider above rates only for calculation of NPV -

First CF = 230 PV = 240*0.995= 228.85

Second CF = 230*(1+0.05/2)# = 235.75 PV = 235.75*0.98 = 231.04

Third CF = 235.75*(1+0.05/2) = 241.64 PV = 241.64*0.95= 229.56

Fourth CF = 241.64*(1+0.05/2) = 247.68 PV = 247.68*0.93 = 230.35

Total = 919.80 Outflow = 900 NPV = 919.80-900= 19.80

# 5% is annualised growth, so 2.5% is growth during g 6m period. I took it as a bond equivalent yield, hence can divide by 2 simply (consistent with rates computed for above bonds). Otherwise, we could do like this -

= 1.05^0.5 = 2.47% for 6 months .So growth would be 2.47% not 2m5% now onwards. This is accurate but I respected simplicity.